Morning Michael,
there is no difference in how RAxML (or other ML programmes) and MrBayes (or other Bayesian programmes) compute support values. The values are simply counted from the samples and reflect the frequencies of a given taxon bipartition (= a branch in your tree) in the tree sample, i.e. the bootstrap sample in case of ML-bootstrapping and non-burned sampled topologies in case of Bayesian inference.
Usually, PP and ML-BS are congruent, although PP can tend to overestimate support and ML-BS can be underestimating (hence the current arbitrary thresholds for good support, BS > 70, PP > 0.95).
A clear signal will always have PP = 1.0 and ML-BS = 100.
When you have a branch/taxon bipartition, a phylogenetic split, where PP >> ML-BS then it usually a signal issue in your underlying matrix: Either
1) a faint signal, very few sites supporting the split, hence, a higher chance that a BS-replicate eliminated those sites but PP remain high, because they do not resample the underlying data and the signal was lost (in this case no topological alternative receives meaningful support with ML-BS), or
2) actual signal conflict between concatenated gene regions (in this case there are two or more alternative receiving ample/subtractive support). For instance, in the perfect case if you have 20% of segregating sites strictly following a topology A and 80% strictly following a topology B, BS-ML will be split 20:80 for both alternatives but PP ~ 1.0, because if 80% of the segragating sites support topology B, there be a very high probability that B is correct and A wrong (when we infer a tree, we use algorithms that work under the assumption that there is only one single-correct tree that produced the entire data set)
To see how your PP and ML-BS relate to each other in total, you can use
a) the bipartition frequency mapping option in RAxML (option -f m; which gives you an x-y plot and computes the Pearson correlation coefficient)
b) investigate the bootstrap replicate sample and Bayesian sample using the consensus network approach implemented in SplitsTree (you can just open the RAxML_bootstrap and .t file, make sure you delete the burnin fraction, with SplitsTree; choose COUNT in the popping up menu to optain edge lengths that are proportionate to the support of the according split; and the updated Phangorn Package for R,
https://peerj.com/preprints/2054; see vignettes for guidelines how to use the new network functions)
To have disparitate PP / ML-BS support for the branch after the root node is not uncommon and - to my experience - shows that you outgroup-inferred root is problematic (BS and PP react differently to ambiguous signals). Particular problematic are outgroups that are very distinct from your ingroup taxa (ingroup-outgroup branching artefacts). You can use RAxML for a quick test:
Step 1: Run an analysis without the outgroups
Step 2: Use the evolutionary placement algorithm implemented in RAxML (-f v) to find the optimal position of each outgroup taxon within the ingroup-only tree (there have been quite a bunch of posts on this and e.g. this paper:
http://dx.doi.org/10.1080/14772000.2014.941037 -- if you have no access, try RG or Academia)
Cheers, Guido